REVIEW FOR QUIZ!
NAME: __________________________
Solve each equation (use the method provided)
1. Square Roots Method
2.
(x + 3)2 + 2 = –10
3.
Complete the Square.
x2 – 8x + 3 = 0
Factor to solve.
x2 – 2x – 15 = 0
4. Quadratic Formula.
6x2 + 2x + 1= 0
DUE TUES/WED
Solve each equation.
5.
y = 2 (x – 3)2 + 8
7.
y = x2 – 14x + 1
(square roots)
(complete the square)
6.
y = 2x2 + x – 10
8.
y = x2 – 2x + 5
(factor and zero prod)
(quadratic formula)
SOLVING: WHICH METHOD SHOULD YOU USE?
Explain why!
1
Equation
x + 4x + 3 = 0
A
Sq. Roots
B
Factor/ZPP
C
Complete Sq.
D
Quad. Form
2
5x2 – 1 = 6
Sq. Roots
Factor/ZPP
Complete Sq.
Quad. Form
3
x2 – 7x + 1 = 0
Sq. Roots
Factor/ZPP
Complete Sq.
Quad. Form
4
x2 + 10x + 4 = 0
Sq. Roots
Factor/ZPP
Complete Sq.
Quad. Form
5
x2 – 14x = 5
Sq. Roots
Factor/ZPP
Complete Sq.
Quad. Form
6
5 – 3x2 = 20
Sq. Roots
Factor/ZPP
Complete Sq.
Quad. Form
7
x2 + x = 10
Sq. Roots
Factor/ZPP
Complete Sq.
Quad. Form
8
x2 – 4x – 12 = 0
Sq. Roots
Factor/ZPP
Complete Sq.
Quad. Form
2
Solve: Choose which method is best =)
1.
y = 2 (x + 2)2 + 24
2.
y = x2 – 6x + 8
3.
y = 2x 2 − 5x −12
4.
y = x2 – 12x + 1
Find the discriminant and determine the number and type of solutions.
Discriminant
5. y = 3x2 – 3x + 2
6. y = x2 – 10x + 1
7. y = x2 – 4x + 4
Number and Type of Solutions
REVIEW PACKET SECTION 1: FACTORING
Factor Completely!
1.
x2
–
7x
+
6
2.
x2
–
100
3.
4x2
+
81
4.
25p2
–
16p
5.
m2
–
10m
+
21
6.
y2
–
3y
–
18
7.
x2
+
7x
+
12
8.
4x2
+
20x
–
24
9.
4x2
+
20x
+
25
10.
16a2
–
49b2
11.
16x2
+
6x
12.
x2
–
2x
+
1
13.
16a2
–
81
14.
3x2
+
3x
–
36
15.
x2
–
8x
+
16
16.
24z2
–
14z
–
5
17.
3m2
–
7m
+
2
18.
3x2
–
75
SECTION 2: SOLVING
Use
the
square
root
method.
1.
5x2
–
7
=
60
2.
x2
+
16
=
0
3.
4.
2(x+3)2
+
12
=
4
Factor
and
use
the
zero
product
property.
5.
(2x
+
8)
(x
–
5)
=
0
6.
x2
–
2x
+
1
=
0
7.
8.
6x2
+
11x
=
10
5x2
+
9
=
134
x2
+
6x
=
0
Complete
the
Square.
9.
x2
–
4x
–
12
=
0
10.
x2
–
2x
–
35
=
0
12.
4x2
–
8x
=
40
Use
the
Quadratic
Formula.
13.
x2
+
5x
–
6
=
0
14.
2x2
–
4x
+
3
=
0
15.
16.
10x2
+
9
=
x
11.
x2
+
6x
=
23
2x2
–
x
–
4
=
2
SECTION 3: GRAPHING
Find the vertex of each quadratic function:
1. f(x) = (x+ 2)2 + 5
(
,
3. f(x) = (x – 1)2
(
,
)
)
5. f(x) = (x + 10) (x – 2)
4. f(x) = 5x2
(
,
)
(
,
)
(
,
)
(
,
)
6. f(x) = x2 + 2x + 5
(
,
)
7. f(x) = 2 (x – 5) (x + 3)
8. f(x) = 2x2 + 8x + 5
(
9.
y
=
–3 (x – 1)2 + 10 10.
y
=
(x + 4)2 + 4
2. f(x) = –2x2 – 3
,
)
Opens
Up
or
Opens
Down
Stretched,
Shrink,
Standard
Opens
Up
or
Opens
Down
Stretched,
Shrink,
Standard
11. Name 3 synonyms for “solution”: _______________, _______________, _______________
Graph.
12.
y = 2 ( x + 5) − 3
2
13.
y = − 1 ( x + 5) ( x − 3)
2
14.
y = x 2 + 4x − 6
Quick Questions. Choose either ANSWER A or ANSWER B.
QUESTION
ANSWER A
What is the form of the function:
1
Intercept Form
y = 2x2 + 3x + 2
What is the form of the function:
2
Vertex Form
y = 2(x + 3)2 – 10
What is the form of the function:
3
Intercept Form
y = – (x + 3) (x – 8)
4
5
6
7
8
9
10
11
12
13
14
15
16
What formula will find the x-coordinate
of the vertex for standard form?
What formula will find the x-coordinate
of the vertex for intercept form?
What is the value of C that would
complete the square: x2 – 4x + C
What is the a-value: y = 2x2 + 5x + 2
What type of polynomial is always
prime?
What method would you use to solve the
equation: y = (x + 3) (2x + 1)
What method would you use to solve the
equation: y = 4x2 + 10
What method would you use to solve the
equation: y = x2 + 10x + 3
The discriminant is 24. How many
solutions are there?
The discriminant is -10. How many
solutions are there?
The discriminant is 0. How many
solutions are there?
The discriminant is -25. What type of
solutions are there?
The discriminant is 4. What type of
solutions are there?
−b
x=
2a
x=
p−q
2
ANSWER B
Standard Form
Intercept Form
Standard Form
⎛b⎞
x =⎜ ⎟
⎝ 2⎠
x=
2
p+q
2
4
16
1
2
A binomial sum of
squares
A trinomial
Zero Product Property
Complete the Square
Square Roots Method
Quadratic Formula
Square Roots Method
Complete the Square
2
1
0
2
1
0
Real
Imaginary
Real
Imaginary
17
How do you find any x-intercept?
Substitute 0 for x
Substitute 0 for y
18
How do you find any y-intercept?
Substitute 0 for x
Substitute 0 for y
19
What is the quadratic formula?
−b b 2 − 4ac
x=
2a
−b ± b 2 − 4ac
x=
2a
20
What calculator function can you use to
find the vertex of a parabola?
2nd Graph
2nd TRACE
WHAT SHOULD YOU DO NEXT?
(when solving with square roots or factoring methods)
1.
3.
5.
2x2 + 8 = 10
2.
(x + 4)2 = 25
A.
Divide both sides by 2.
A.
Distribute the square.
B.
Isolate x2.
B.
FOIL.
C.
Square root both sides.
C.
Square root both sides.
x2 – 25x = 0
4.
x2 + 5x + 4 = 0
A.
Factor into (x + 5) (x – 5).
A.
Square root both sides.
B.
Add 25x to both sides.
B.
Subtract 4 from both sides.
C.
Factor out x.
C.
Factor the trinomial.
x2 + 3x = 10
6.
(3x + 1) (x + 4) = 0
A.
Square root both sides.
A.
Set each factor equal to 0.
B.
Subtract 3x from both sides.
B.
FOIL.
C.
Subtract 10 from both sides.
C.
Combine like terms.
7.
2
2x + 7x + 3
WHAT SHOULD YOU DO NEXT in order to factor?
8.
9x2 – 30x + 25
A.
List pairs of factors of 3.
A.
Try (3x – 5)2 and check it.
B.
Multiply 2 and 3.
B.
Multiply 9 and 25.
C.
Factor out x.
C.
Set it equal to 0 and solve.
WRITE THE NEXT STEP ONLY!
1.
(x + 5)2 = –49
2.
x2 – 9x = 0
3.
2x2 + 4 = 8
4.
4x2 – 81 = 0
5.
x–2= ± 3
6.
x + 1 = ±6i
7.
Complete the square.
8.
Complete the square.
x2 – 6x + 10 = 0
9.
Complete the square.
x2 + 8x = 3
10.
x2 + 10x + 25 = 6
11.
Quadratic Formula.
x=
2 ± 9 − 2(−2)(−4)
4
Quadratic Formula.
x2 + 8x = 3
12.
Quadratic Formula.
x=
−10 ± 6i 2
4
Unit 4 QUADRATICS Summary Sheet
SECTION 1: FACTORING
1.
Put the polynomial in order of decreasing degree
(standard form).
10 + 7x + x2
2.
Factor out the GCF (include any variables!)
4x2 + 14x
All Types
Binomial
A2 – B2
If it is a difference of squares, factor into conjugates.
Formula: ___________________________________
x2 – 100
Binomial
A2 + B2
If it is a sum of squares, the binomial is PRIME.
x2 + 100
If A = 1,
1. List the pairs of factors of C.
2. Find a pair that has a sum/difference of the target #.
3. Write the two binomials.
x2 + 7x + 12
If A = 1,
1. Multiply A and C and list pairs of factors.
2. Find a pair that has a sum/difference of the target #.
3. Factor by grouping.
(or factor by trial and error)
2x2 – 3x – 20
Trinomial
x2 + Bx + C
Trinomial
x2 + Bx + C
Perfect
Square
Trinomial
1.
2.
3.
4x2 + 28x + 49
If the first and last terms are perfect squares:
Try writing it as a binomial squared.
CHECK that the middle term works!!
SECTION 2: GRAPHING
A quadratic function is a function with 2 as the highest degree (exponent)
Vertex Form
Intercept Form
Standard Form
y = a (x − h) + k
y = a(x − p)(x − q)
y = ax 2 + bx + c
2
Vertex: (h, k)
1. a > 0: opens up
a < 0: opens down
2. a < -1 or a > 1: stretched
-1 < a < 1: compressed
3. Use the squares chart to find
other points on the graph.
⎛
Vertex: ⎜
⎝
p+q
, f
2
( ) ⎞⎟⎠
p+q
2
⎛ −b
Vertex: ⎜
, f
⎝ 2a
( ) ⎞⎟⎠
−b
2a
1. Find the x-coordinate of the
vertex.
1. Find the x-coordinate of the
vertex.
2. Substitute it into the function
to find the y-coordinate of the
vertex.
2. Substitute it into the function
to find the y-coordinate of the
vertex.
3. Use the chart to find other
points on the graph.
3. Use the chart to find other
points on the graph.
SECTION 3: SOLVING
1.
Square Roots.
Use When:
2.
An equation has an x2 or (x + c)2
(but does not have an x)
Factor and Zero Product Property.
Use When:
The equation is factorable.
1.
Make sure the equation is in the form:
ax2 + bx + c = 0
2
1.
Isolate the x .
2.
Square root both sides.
2.
Factor completely!
3.
Simplify (including the square root!)
3.
Set each factor equal to 0.
4.
Don’t forget the ± sign!
4.
Solve.
5.
Write the solutions together: x = ____, ____
4.
Quadratic Formula.
3.
Complete the Square.
Use When:
1.
The trinomial is not factorable.
A=1 and B is even.
Make sure the equation is in the form:
Ax2 + Bx = C
2
2.
⎛ B⎞
Use the formula ⎜ ⎟ to determine C.
⎝ 2⎠
3.
Add C to both sides.
4.
Factor the left side of the equation into a
binomial squared.
5.
Take the square root of both sides (don’t
forget ± )
6.
Isolate the x.
Use When:
The other methods do not apply.
1. Put the equation into standard form:
Ax2 + Bx + C = 0
2. Find A, B, C.
3. Substitute A, B, and C into the quadratic
formula. Use parentheses!
4. Simplify completely!
Quadratic Formula: x =
−b ± b 2 − 4ac
2a
Discriminant : b2 – 4ac
If negative = 2 imaginary solutions
If 0 = one real number solution
If positive = 2 real number solutions
Recall, i =
−1
Quadratic Equations Methods
Name: _______________________________________________
I.
What makes an equation a quadratic equation?
II.
There are four methods. List them!
Period: ________
A.
B.
C.
D.
III.
How can you determine which method to use?
A.
USE SQUARE ROOTS METHOD IF:
If the equation has ______________________ OR ______________________ ,
(and no ____________)
B.
FACTOR AND USE THE ZERO PRODUCT PROPERTY IF:
If the equation has ______________________ AND ______________________ ,
IF THE FIRST TWO METHODS DON’T WORK, CHOOSE BETWEEN THESE TWO:
C.
COMPLETE THE SQUARE IF:
A=1
D.
AND
the middle term is _________________.
USE THE QUADRATIC FORMULA IF:
The middle term is _________________.
1.
2.
x2 + 6x + 5 = 0
4(x+2)2 + 100 = 14
3.
4.
x2 – 49 = 0
x2 + 8x + 1 = 10
5.
6.
-2 = 2x2 + 8
x2 + 3x + 1 = 0
7.
8.
4x2 + 12x + 9 = 0
x2 + 6x + 3 = 0
9.
10.
x2 – 36x = 0
x2 + 100 = 0
11.
12.
x2 + 8x + 16 = 3
x2 + 11x = 4
1. x2 + 6x + 5 = 0
2. 4(x+2)2 + 100 = 14
Factor/ZPP
Sq. Roots Method
3. x2 – 49 = 0
4. x2 + 8x + 1 = 10
Factor/ZPP
Complete the Square
5. -2 = 2x2 + 8
6. x2 + 3x + 1 = 0
Sq. Roots Method
Quadratic Formula
7. 4x2 + 12x + 9 = 0
8. x2 + 6x + 3 = 0
Factor/ZPP
Complete the Square
9. x2 – 36x = 0
10. x2 + 100 = 0
Factor/ZPP
Sq. Roots Method
11. x2 + 8x + 16 = 3
12. x2 + 11x = 4
Complete the Square
Quadratic Formula
QUADRATIC EQUATIONS SORT MAT
Sq. Roots Method
Factor/Zero Product Property
Complete the Square
Quadratic Formula
QUADRATIC EQUATIONS SORT MAT
Sq. Roots Method
Factor/Zero Product Property
2, 5, 10
1, 3, 7, 9
Complete the Square
Quadratic Formula
4, 8, 11
6, 12